During the same period that "Near East" moved towards
obsolescence among the British, it became the preferred term of
the U.S. Department of State. In 1944, the State Department had
an Office of Near Eastern and African Affairs, subdivided into three
divisions: Near Eastern, Middle Eastern, and African. The region
covered by the Division of Middle Eastern Affairs corresponded to
the earliest British usage of "Middle East." The core
of the region usually referred to today as the Middle East (Turkey,
the Levant, Arabia, Egypt) was covered by the Division of Near Eastern
Affairs.

Source:U.S. Government
Manual, Summer 1944, 191.

"Middle East"
AfghanistanBurmaCeylonIndiaIranNepal

"Near East"
EgyptGreeceIraqLebanonPalestine
Saudi
Arabia and other countries of the Arabian peninsulaSyriaTransjordanTurkey